Place an atom in a magnetic field, and (as Zeeman discovered) its spectral
lines will split. Pais's book Inward Bound gives fascinating details
of the experimental history.

For some atoms, spectral lines split in three under a magnetic field. This
is known as the normal Zeeman effect. For other atoms, the spectrum
displays a more complex pattern of splittings, known as the anomalous Zeeman effect. When the magnetic field becomes strong enough,
however, some lines merge back together, and the anomalous Zeeman splitting
coalesces smoothly into a normal splitting-- this is called the Paschen-Back effect. Young Heisenberg wrestled with the anomalous Zeeman
effect, as a student in Sommerfeld's seminar on atomic spectra (see
Cassidy's biography Uncertainty.)

The full explanation of all this phenomena presented quite a puzzle to the
quantum pioneers. Even today, fitting all the pieces together can be
confusing. In the next few sections I will try to assemble this puzzle.

I treat the Zeeman effect in such gory detail for two reasons. First,
historical interest: the Zeeman effect gave the first hints of the
complexities of quantum spin. Many of the strange ``spin facts'' I stated
earlier were first discovered empirically, from poring over spectra and
noticing patterns.

Second, ``gestalt'': the very complexity of the Zeeman effect means that any
explanation of it must bring together several aspects of quantum mechanics.

We can get started by comparing figures 1 and 2. We
see that spectral lines will split if energy levels move up or down (i.e.,
if the energy levels of quantum states change). To progress further, one
needs a formula for the energy levels of the states of an atom-- or
rather, a formula for the change in energy level due to the magnetic field.

Say we write
, where is the energy with no
magnetic field, and is the energy due to the magnetic field.
(I will add quantum-number subscripts (like we had with ) in a
moment.) Here is a formula for that accounts for the normal
Zeeman effect:

(1)

where , , , and are quantum numbers that I will discuss
below, and is the strength of the magnetic field. Here is a formula
that accounts for the anomalous Zeeman effect:

(2)

where and are another pair of quantum numbers to be discussed
below. The complicated fraction built out of , , and on the
right hand side is known as the Landé -factor.

So there are four pieces to the puzzle:

What is the meaning of the six quantum numbers , , , ,
, and ?

How do the ``normal'' and ``anomalous'' formulas (formulas 1
and 2) account for the details of the Zeeman splitting?

Where do the normal and anomalous Zeeman formulas come from?

How does the anomalous Zeeman formula turn into
the normal Zeeman formula in the Paschen-Back limit of strong magnetic
fields?